### Mike Cotton

The CENTROID (G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

The ORTHOCENTER (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side. (Note: The foot of the perpendicular may be on the extension of the side of the triangle.)

The CIRCUMCENTER (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle.

The INCENTER (I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides.

After performing several investigations it was found that if the triangle is positioned such that one side is fixed horizontally (segment YZ), and the vertex that is not on the horozontal side (vertex X) is moved horizontally (on line Base,either above of below the fixed side) the orthocenter traces a parabola. See the figure below.

Figure1

What is interesting about this figure is that the distance from the focus to the directrix (FocusM = 1.000 in.) for the parabola is half the lnegth of the segment ZX
(XZ = 2.00 in.) when the vertex Z is a right angle. That is, there is a right triangle with the right angle at Z. The distance from Z to X' represents length of segment ZX when there is a right angle at vertex Z. The example above is where segment ZX is less than half of segment YZ.

Below are examples, one where segment ZX = 1/2 segment YZ (Figure 2), and the other is where segment ZX is greater then one-half the length of segment YZ (Figure 3).

Figure 2

Figure 3

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